Input TRS: 1: minus(x,|0|()) -> x 2: minus(s(x),s(y)) -> minus(x,y) 3: div(|0|(),s(y)) -> |0|() 4: div(s(x),s(y)) -> s(div(minus(x,y),s(y))) 5: divL(x,nil()) -> x 6: divL(x,cons(y,xs())) -> divL(div(x,y),xs()) 7: app(nil(),y) -> y 8: app(cons(n,x),y) -> cons(n,app(x,y)) 9: reverse(nil()) -> nil() 10: reverse(cons(n,x)) -> app(reverse(x),cons(n,nil())) 11: shuffle(nil()) -> nil() 12: shuffle(cons(n,x)) -> cons(n,shuffle(reverse(x))) 13: concat(leaf(),y) -> y 14: concat(cons(u,v),y) -> cons(u,concat(v,y)) 15: less_leaves(x,leaf()) -> false() 16: less_leaves(leaf(),cons(w,z)) -> true() 17: less_leaves(cons(u,v),cons(w,z)) -> less_leaves(concat(u,v),concat(w,z)) 18: divL(z,cons(x,xs())) ->= divL(z,consSwap(x,xs())) 19: shuffle(cons(x,xs())) ->= shuffle(consSwap(x,xs())) 20: consSwap(x,xs()) ->= cons(x,xs()) 21: consSwap(x,cons(y,xs())) ->= cons(y,consSwap(x,xs())) Number of strict rules: 17 Direct Order(PosReal,>,Poly) ... failed. Freezing ... failed. Dependency Pairs: #1: #minus(s(x),s(y)) -> #minus(x,y) #2: #divL(x,cons(y,xs())) -> #divL(div(x,y),xs()) #3: #divL(x,cons(y,xs())) -> #div(x,y) #4: #shuffle(cons(n,x)) -> #shuffle(reverse(x)) #5: #shuffle(cons(n,x)) -> #reverse(x) #6: #concat(cons(u,v),y) -> #concat(v,y) #7: #reverse(cons(n,x)) -> #app(reverse(x),cons(n,nil())) #8: #reverse(cons(n,x)) -> #reverse(x) #9: #less_leaves(cons(u,v),cons(w,z)) -> #less_leaves(concat(u,v),concat(w,z)) #10: #less_leaves(cons(u,v),cons(w,z)) -> #concat(u,v) #11: #less_leaves(cons(u,v),cons(w,z)) -> #concat(w,z) #12: #shuffle(cons(x,xs())) ->? #shuffle(consSwap(x,xs())) #13: #app(cons(n,x),y) -> #app(x,y) #14: #div(s(x),s(y)) -> #div(minus(x,y),s(y)) #15: #div(s(x),s(y)) -> #minus(x,y) #16: #divL(z,cons(x,xs())) ->? #divL(z,consSwap(x,xs())) Number of SCCs: 8, DPs: 9, edges: 11 SCC { #8 } Removing DPs: Order(PosReal,>,Sum)... succeeded. |0|() weight: 0 #div(x1,x2) weight: 0 s(x1) weight: 0 #reverse(x1) weight: x1 minus(x1,x2) weight: 0 #less_leaves(x1,x2) weight: 0 #concat(x1,x2) weight: 0 false() weight: 0 div(x1,x2) weight: 0 reverse(x1) weight: 0 true() weight: 0 shuffle(x1) weight: 0 concat(x1,x2) weight: 0 consSwap(x1,x2) weight: 0 less_leaves(x1,x2) weight: 0 nil() weight: 0 #app(x1,x2) weight: 0 divL(x1,x2) weight: 0 #minus(x1,x2) weight: 0 #shuffle(x1) weight: 0 cons(x1,x2) weight: (/ 1 2) + x2 xs() weight: 0 leaf() weight: 0 #divL(x1,x2) weight: 0 app(x1,x2) weight: 0 Usable rules: { } Removed DPs: #8 Number of SCCs: 7, DPs: 8, edges: 10 SCC { #13 } Removing DPs: Order(PosReal,>,Sum)... succeeded. |0|() weight: 0 #div(x1,x2) weight: 0 s(x1) weight: 0 #reverse(x1) weight: 0 minus(x1,x2) weight: 0 #less_leaves(x1,x2) weight: 0 #concat(x1,x2) weight: 0 false() weight: 0 div(x1,x2) weight: 0 reverse(x1) weight: 0 true() weight: 0 shuffle(x1) weight: 0 concat(x1,x2) weight: 0 consSwap(x1,x2) weight: 0 less_leaves(x1,x2) weight: 0 nil() weight: 0 #app(x1,x2) weight: x1 divL(x1,x2) weight: 0 #minus(x1,x2) weight: 0 #shuffle(x1) weight: 0 cons(x1,x2) weight: (/ 1 2) + x2 xs() weight: 0 leaf() weight: 0 #divL(x1,x2) weight: 0 app(x1,x2) weight: 0 Usable rules: { } Removed DPs: #13 Number of SCCs: 6, DPs: 7, edges: 9 SCC { #6 } Removing DPs: Order(PosReal,>,Sum)... succeeded. |0|() weight: 0 #div(x1,x2) weight: 0 s(x1) weight: 0 #reverse(x1) weight: 0 minus(x1,x2) weight: 0 #less_leaves(x1,x2) weight: 0 #concat(x1,x2) weight: x1 false() weight: 0 div(x1,x2) weight: 0 reverse(x1) weight: 0 true() weight: 0 shuffle(x1) weight: 0 concat(x1,x2) weight: 0 consSwap(x1,x2) weight: 0 less_leaves(x1,x2) weight: 0 nil() weight: 0 #app(x1,x2) weight: 0 divL(x1,x2) weight: 0 #minus(x1,x2) weight: 0 #shuffle(x1) weight: 0 cons(x1,x2) weight: (/ 1 2) + x2 xs() weight: 0 leaf() weight: 0 #divL(x1,x2) weight: 0 app(x1,x2) weight: 0 Usable rules: { } Removed DPs: #6 Number of SCCs: 5, DPs: 6, edges: 8 SCC { #1 } Removing DPs: Order(PosReal,>,Sum)... succeeded. |0|() weight: 0 #div(x1,x2) weight: 0 s(x1) weight: (/ 1 2) + x1 #reverse(x1) weight: 0 minus(x1,x2) weight: 0 #less_leaves(x1,x2) weight: 0 #concat(x1,x2) weight: 0 false() weight: 0 div(x1,x2) weight: 0 reverse(x1) weight: 0 true() weight: 0 shuffle(x1) weight: 0 concat(x1,x2) weight: 0 consSwap(x1,x2) weight: 0 less_leaves(x1,x2) weight: 0 nil() weight: 0 #app(x1,x2) weight: 0 divL(x1,x2) weight: 0 #minus(x1,x2) weight: x2 #shuffle(x1) weight: 0 cons(x1,x2) weight: (/ 1 2) xs() weight: 0 leaf() weight: 0 #divL(x1,x2) weight: 0 app(x1,x2) weight: 0 Usable rules: { } Removed DPs: #1 Number of SCCs: 4, DPs: 5, edges: 7 SCC { #16 } Removing DPs: Order(PosReal,>,Sum)... Order(PosReal,>,Max)... QLPOpS... Order(PosReal,>,MaxSum)... QWPOpS(PosReal,>,MaxSum)... Order(PosReal,>,Sum-Sum; PosReal,≥,Sum-Sum)... Order(PosReal,>,Sum-Sum; NegReal,≥,Sum)... Order(PosReal,>,MaxSum-Sum; NegReal,≥,Sum)... failed. Removing edges: failed. Finding a loop... failed. MAYBE